Subgroups — D₃×ΓL(1,𝔽₈)×D₇₁ #31822074

Order 17892.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₂₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×ΓL(1,𝔽₈)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×ΓL(1,𝔽₈)×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₇×D₇₁	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×ΓL(1,𝔽₈)×D₇₁	—	3	1	no	maximal
6	C₃×ΓL(1,𝔽₈)×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
7	ΓL(1,𝔽₈)×D₇₁	—	2	1	no	—
8	ΓL(1,𝔽₈)×D₇₁	—	3	1	no	—
9	ΓL(1,𝔽₈)×D₇₁	D₃	1	1	yes	characteristic, direct product factor
10	C₇×D₂₁₃	C₂×C₃	1	1	yes	characteristic, complemented
11	D₃×C₇×C₇₁	C₂×C₃	1	1	yes	characteristic, complemented
12	C₃×C₇×D₇₁	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×ΓL(1,𝔽₈)×C₇₁	—	3	1	no	—
14	C₃×D₃×D₇₁	—	7	1	no	maximal, Hall
15	C₂×C₇×D₇₁	—	3	1	no	Hall
16	C₃×C₇×C₇₁	C₂²×C₃	1	1	yes	characteristic, complemented
17	ΓL(1,𝔽₈)×C₇₁	—	2	1	no	—
18	ΓL(1,𝔽₈)×C₇₁	C₂×D₃	1	1	yes	characteristic, complemented
19	C₃×D₂₁₃	—	7	1	no	—
20	C₃×D₃×C₇₁	—	7	1	no	—
21	C₃²×D₇₁	—	7	1	no	—
22	C₂×C₇×C₇₁	—	3	1	no	—
23	C₇×D₇₁	—	3	1	no	—
24	C₇×D₇₁	C₃×D₃	1	1	yes	characteristic, complemented
25	C₂×C₃×D₇₁	—	21	1	no	—
26	D₃×D₇₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
27	C₃²×C₇₁	—	7	1	no	Hall
28	C₇×C₇₁	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₇₁	—	21	1	no	—
30	D₂₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	C₃×D₇₁	—	7	1	no	—
32	C₃×D₇₁	—	14	1	no	—
33	C₃×D₇₁	—	21	1	no	—
34	C₃×D₇₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
35	D₃×C₇₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
36	C₂×D₇₁	—	3	1	no	Hall
37	C₂×D₃×ΓL(1,𝔽₈)	—	71	1	no	maximal, Hall
38	C₃×C₇₁	—	7	1	no	—
39	C₃×C₇₁	—	14	1	no	—
40	C₃×C₇₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
41	C₂×C₇₁	—	3	1	no	—
42	D₇₁	—	3	1	no	—
43	D₇₁	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
44	C₂×C₃×ΓL(1,𝔽₈)	—	71	1	no	—
45	D₃×ΓL(1,𝔽₈)	—	71	1	no	—
46	D₃×ΓL(1,𝔽₈)	D₇₁	1	1	yes	characteristic, direct product factor
47	C₂×D₃×C₇	—	71	1	no	—
48	C₂²×ΓL(1,𝔽₈)	—	213	1	no	—
49	C₇₁	C₂×D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
50	C₃×ΓL(1,𝔽₈)	C₂×D₇₁	1	1	yes	characteristic, complemented, Hall
51	C₂×C₃×C₇	—	71	1	no	—
52	D₃×C₇	—	71	1	no	—
53	D₃×C₇	C₃×D₇₁	1	1	yes	characteristic, complemented
54	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
55	C₂×ΓL(1,𝔽₈)	—	71	1	no	—
56	C₂×ΓL(1,𝔽₈)	—	142	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	213	1	no	—
58	C₂×C₃×D₃	—	497	1	no	Hall
59	C₂²×C₇	—	213	1	no	Hall
60	C₃×C₇	C₂×C₃×D₇₁	1	1	yes	characteristic, complemented
61	ΓL(1,𝔽₈)	—	2	1	no	—
62	ΓL(1,𝔽₈)	D₃×D₇₁	1	1	yes	characteristic, direct product factor
63	C₂×C₃²	—	497	1	no	—
64	C₃×D₃	—	7	1	no	—
65	C₃×D₃	—	497	1	no	—
66	C₂×C₇	—	3	1	no	—
67	C₂×C₇	—	71	1	no	—
68	C₂×C₇	—	213	1	no	—
69	C₂²×C₃	—	1491	1	no	—
70	C₂×D₃	—	71	1	no	—
71	C₃²	—	7	1	no	Sylow
72	C₇	C₃×D₃×D₇₁	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	21	1	no	—
74	C₂×C₃	—	71	1	no	—
75	C₂×C₃	—	1491	1	no	—
76	C₂×C₃	—	994	1	no	—
77	C₂×C₃	—	497	1	no	—
78	D₃	—	71	1	no	—
79	D₃	ΓL(1,𝔽₈)×D₇₁	1	1	yes	characteristic, direct product factor
80	C₂²	—	213	1	no	Sylow
81	C₃	—	7	1	no	—
82	C₃	—	14	1	no	—
83	C₃	C₂×ΓL(1,𝔽₈)×D₇₁	1	1	yes	characteristic, complemented
84	C₂	—	3	1	no	—
85	C₂	—	71	1	no	—
86	C₂	—	213	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₇₁	C₂²×C₃
LCS	G ▷ C₃×C₇×C₇₁
UCS	C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₂₁₃	C₂	semidirect
2. D₃×ΓL(1,𝔽₈)×C₇₁	C₂	semidirect
3. C₃×ΓL(1,𝔽₈)×D₇₁	C₂	semidirect
4. D₃×C₇×D₇₁	C₃	semidirect
6. C₃×ΓL(1,𝔽₈)×C₇₁	C₂²	semidirect
9. ΓL(1,𝔽₈)×D₇₁	D₃	direct (N × H)
10. C₇×D₂₁₃	C₂×C₃	semidirect
11. D₃×C₇×C₇₁	C₂×C₃	semidirect
12. C₃×C₇×D₇₁	C₂×C₃	semidirect
16. C₃×C₇×C₇₁	C₂²×C₃	semidirect
18. ΓL(1,𝔽₈)×C₇₁	C₂×D₃	semidirect
24. C₇×D₇₁	C₃×D₃	semidirect
26. D₃×D₇₁	ΓL(1,𝔽₈)	direct (N × H)
28. C₇×C₇₁	C₂×C₃×D₃	semidirect
30. D₂₁₃	C₂×ΓL(1,𝔽₈)	semidirect
34. C₃×D₇₁	C₂×ΓL(1,𝔽₈)	semidirect
35. D₃×C₇₁	C₂×ΓL(1,𝔽₈)	semidirect
40. C₃×C₇₁	C₂²×ΓL(1,𝔽₈)	semidirect
43. D₇₁	D₃×ΓL(1,𝔽₈)	direct (N × H)
46. D₃×ΓL(1,𝔽₈)	D₇₁	direct (N × H)
49. C₇₁	C₂×D₃×ΓL(1,𝔽₈)	semidirect
50. C₃×ΓL(1,𝔽₈)	C₂×D₇₁	semidirect
53. D₃×C₇	C₃×D₇₁	semidirect
60. C₃×C₇	C₂×C₃×D₇₁	semidirect
62. ΓL(1,𝔽₈)	D₃×D₇₁	direct (N × H)
72. C₇	C₃×D₃×D₇₁	semidirect
79. D₃	ΓL(1,𝔽₈)×D₇₁	direct (N × H)
83. C₃	C₂×ΓL(1,𝔽₈)×D₇₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
ΓL(1,𝔽₈)×D₂₁₃	2
D₃×ΓL(1,𝔽₈)×C₇₁	2
C₃×ΓL(1,𝔽₈)×D₇₁	2
D₃×C₇×D₇₁	3
C₂×ΓL(1,𝔽₈)×D₇₁	3
C₃×D₃×D₇₁	7
C₂×D₃×ΓL(1,𝔽₈)	71

Subgroups — D3×ΓL(1,𝔽8)×D71 #31822074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×ΓL(1,𝔽₈)×D₇₁ #31822074